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I was wondering if someone could explain to me how the distribution of mass affects angular momentum. I have tried to do some research but had no luck anywhere.

Sherwood
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2 Answers2

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Have a look at the Wikipedia article on moment of inertia as a starting point. In particular, the definition section includes an explanation of how mass distributions come into play:

For a simple pendulum, this definition yields a formula for the moment of inertia $I$ in terms of the mass $m$ of the pendulum and its distance $r$ from the pivot point as, \begin{equation} I = m r^2 . \end{equation} Thus, moment of inertia of the pendulum depends on both the mass $m$ of a body and its geometry, or shape, as defined by the distance $r$ to the axis of rotation.

This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses $\mathrm{d}m$ each multiplied by the square of its perpendicular distance $r$ to an axis $k$. An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.

hiccups
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The distribution of mass affects the angular momentum through the moment of inertia (usually $I$).

The angular momentum is given by: $\vec L = I \vec \omega $ where $\vec \omega$ is the angular velocity.

Let's assume we're talking about a rigid body spinning around a single axis. If the mass is concentrated near the axis, then it has a relatively low moment of inertia. The further away the mass is from the axis, the higher the moment of inertia.

For a point mass $m$ a distance $r$ away from the axis of rotation, the moment of inertia is $I=mr^2$. Here's the moment of inertia for some common objects:

A solid sphere: $I=\frac{2}{5}mr^2$

A hollow sphere: $I=\frac{2}{3}mr^2$

The coefficient in front comes from the shape of the mass. For a hollow sphere, the mass is further from the axis of rotation, so it has a higher $I$ for the same $m$.

A really great example is a figure skater. The start out a spin slowly with their arms out and then pull them in to lower their moment of inertia (and by conservation of angular momentum) increase their angular velocity. description here

Wikipedia: Moment of inertia

taciteloquence
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